Evidently, the particular ambiguity set \(\mathcal {U}_T(\hat{\mathbb {P}}_T)\subseteq \mathcal {P}\)  considered will ultimately determine the statistical properties and computational challenges of the associated robust data-driven formulation.
Much of the early literature [1]}, [2]}, [3]}, [4]} focused on ambiguity sets consisting of probability measures sharing certain given moments and shape constraints.
More recent approaches [5]} however consider ambiguity sets which are based on a statistical distance instead. For instance, [6]}, [7]} propose to consider ambiguity sets consisting of all distributions at distance at most \(r\)  to the empirical distribution in the Wasserstein metric.
